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A limit theorem in some dynamic fuzzy systems. (English) Zbl 0796.93076

Studied are limit properties of a fuzzy dynamical system (model) governed by the formula \({\mathbf x}_{k+1}= {\mathbf x}_ k\circ {\mathcal J}\) i.e., \({\mathbf x}_{k+1}(y)=\sup_{x\in{\mathbf X}}(\text{min }({\mathbf x}_ k(x), {\mathcal J}(x,y))\), where \({\mathbf x}_ k\) and \({\mathbf x}_{k+1}\) are successive fuzzy states of the system being defined in a compact metric space \({\mathbf X}\) while \(\mathcal J\) denotes the transition relation of the system. The main results of the paper concern sufficient conditions of convergence of the fuzzy states (with this property expressed in the Hausdorff metric) and provide a characterization of the limits of the series (sequences) of the fuzzy states of the model. An illustrative numerical example is also included.

MSC:

93C42 Fuzzy control/observation systems
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